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IIT JAM 2006 : Mathematics

6 pages, 32 questions, 7 questions with responses, 8 total responses,

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JAM 2006 MATHEMATICS TEST PAPER 1 NOTATIONS USED : Z: The set of all real numbers The set of all integers IMPORTANT NOTE FOR CANDIDATES Objective Part: Attempt ALL the objective questions (Questions 1-15). Each of these questions carries six marks. Each incorrect answer carries minus two. Write the answers to the objective questions in the Answer Table for Objective Questions provided on page 7 only. Subjective Part: Attempt ALL subjective questions (Questions 16-29). Each of these questions carries fifteen marks. 1. 2n +1 + 3n +1 equals n 2 n + 3n lim (A) (B) (C) (D) 2. Let f ( x) = ( x 2)17 ( x + 5) 24 . Then (A) (B) (C) (D) 3. 4. 3 2 1 0 does not have a critical point at 2 has a minimum at 2 has a maximum at 2 has neither a minimum nor a maximum at 2 f f f f f f y Let f ( x, y ) = x 5 y 2 tan 1 . Then x + y equals x y x (A) 2 f (B) 3 f (C) 5 f (D) 7 f Let G be the set of all irrational numbers. The interior and the closure of G are denoted by G 0 and G , respectively. Then (A) G0 = , (B) (C) G = , G= G0 = , G = (D) G0 = G , G = G =G 0 1 cos x 5. Let f ( x) = 2 e t dt. Then f ( / 4) equals sin x (A) (B) (C) (D) 6. 1/ e 2/e 2/e 1/ e Let C be the circle x 2 + y 2 = 1 taken in the anti-clockwise sense. Then the value of the integral 3 22 2 xy + y dx + 3x y + 2 x dy ( ) ( ) C equals (A) (B) (C) (D) 7. 1 /2 0 Let r be the distance of a point P( x, y, z ) from the origin O. Then r is a vector (A) orthogonal to OP (B) normal to the level surface of r at P (C) normal to the surface of revolution generated by OP about x-axis (D) normal to the surface of revolution generated by OP about y-axis 8. Let T : 3 3 be defined by T ( x1 , x2 , x3 ) = ( x1 x2 , x1 x2 , 0) . If N(T) and R(T) denote the null space and the range space of T respectively, then (A) (B) (C) (D) 9. dim N (T ) = 2 dim R(T ) = 2 R(T ) = N (T ) N (T ) R(T ) r Let S be a closed surface for which r . n d = 1 . Then the volume enclosed by the S surface is (A) (B) (C) (D) 1 1/3 2/3 3 2 10. If ( c1 + c2 ln x ) / x is the general solution of the differential equation d2y dy x + kx + y = 0, 2 dx dx 2 x>0, then k equals (A) (B) (C) (D) 11. If A and B are 3 3 real matrices such that rank ( AB) =1, then rank ( BA) cannot be (A) (B) (C) (D) 12. linear and of first order linear and of second order nonlinear and of first order nonlinear and of second order Let G be a group of order 7 and ( x) = x 4 , x G . Then is (A) (B) (C) (D) 14. 0 1 2 3 The differential equation representing the family of circles touching y-axis at the origin is (A) (B) (C) (D) 13. 3 3 2 1 not one one not onto not a homomorphism one one, onto and a homomorphism Let R be the ring of all 2 2 matrices with integer entries. Which of the following subsets of R is an integral domain? (A) (B) (C) (D) 0 y x 0 x 0 x y x : x, y Z 0 0 : x, y Z y 0 : x Z x y : x, y , z Z z 3 15. Let f n ( x) = n sin 2 n +1 x cos x. Then the value of /2 lim n ( lim f /2 f n ( x)dx 0 n 0 n ) ( x) dx is (A) 1 / 2 (B) 0 (C) 1 / 2 (D) 16. (a) Test the convergence of the series nn n n =1 n ! 3 (6) (b) Show that ln (1 + cos x ) ln 2 for 0 x / 2. x2 4 (9) 17. Find the critical points of the function f ( x, y ) = x3 + y 2 12 x 6 y + 40. Test each of these for maximum and minimum. 18. (a) Evaluate xe y2 (15) dx dy, where R is the region bounded by the lines x = 0, y = 1 and the R parabola y = x 2 . (6) (b) Find the volume of the solid bounded above by the surface z = 1 x 2 y 2 and below by the plane z = 0 . (9) 19. Evaluate the surface integral x (12 y y 4 + z 2 ) d , S where the surface S is represented in the form z = y 2 , 0 x 1, 0 y 1 . 20. Using the change of variables, evaluate (15) xy dx dy , where the region R is bounded by the R curves xy = 1, xy = 3, y = 3x and y = 5 x in the first quadrant. (15) 21. (a) Let u and v be the eigenvectors of A corresponding to the eigenvalues 1 and 3 (6) respectively. Prove that u + v is not an eigenvector of A. (b) Let A and B be real matrices such that the sum of each row of A is 1 and the sum of (9) each row of B is 2. Then show that 2 is an eigenvalue of AB. 4 22. Suppose W1 and W2 are subspaces of { (1, 0,1, 0 ) , ( 3, 0,1, 0 ) } respectively. Find a W1 + W2 containing { (1, 0,1, 0 ) , ( 3, 0,1, 0 ) } . 4 spanned by { (1, 2,3, 4 ) , ( 2,1,1, 2 ) } and basis of W1 I W2 . Also find a basis of (15) 23. Determine y0 such that the solution of the differential equation y y = 1 e x , y (0) = y0 has a finite limit as x . (15) 24. Let ( x, y, z ) = e x sin y . Evaluate the surface integral S d , where S is the surface n is the directional derivative of in the of the cube 0 x 1, 0 y 1, 0 z 1 and n direction of the unit outward normal to S . Verify the divergence theorem. (15) 25. Let y = f ( x) be a twice continuously differentiable function on (0, ) satisfying f (1) = 1 and f ( x) = 1 2 1 f , x > 0. x Form the second order differential equation satisfied by y = f ( x ) , and obtain its solution satisfying the given conditions. (15) a b 26. Let G = : a, b, c, d Z be the group under matrix addition and H be the c d subgroup of G consisting of matrices with even entries. Find the order of the quotient group G / H . (15) 27. Let x2 f ( x) = x 0 x 1 x > 1. Show that f is uniformly continuous on [0, ). (15) x 28. Find M n = max , and hence prove that the series 3) x 0 n (1 + nx x n 1 + n x3 n =1 ( ) is uniformly convergent on [0, ). (15) 29. Let R be the ring of polynomials with real coefficients under polynomial addition and polynomial multiplication. Suppose I ={ p R : sum of the coefficients of p is zero}. Prove that I is a maximal ideal of R. (15) 5

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